561 research outputs found
The triangulated categories of framed bispectra and framed motives
An alternative approach to the classical Morel-Voevodsky stable motivic
homotopy theory is suggested. The triangulated category of framed
bispectra and effective framed bispectra
are introduced in the paper. Both triangulated
categories only use Nisnevich local equivalences and have nothing to do with
any kind of motivic equivalences. It is shown that and
recover the classical Morel-Voevodsky triangulated
categories of bispectra and effective bispectra
respectively.
We also recover and as the triangulated category of
framed motivic spectral functors and the
triangulated category of framed motives respectively
constructed in the paper
Reconstructing projective schemes from Serre subcategories
Given a positively graded commutative coherent ring A which is finitely
generated as an A_0-algebra, a bijection between the tensor Serre subcategories
of qgr A and the set of all subsets Y\subseteq Proj A of the form
Y=\bigcup_{i\in\Omega}Y_i with quasi-compact open complement Proj A\Y_i for all
i\in\Omega is established. To construct this correspondence, properties of the
Ziegler and Zariski topologies on the set of isomorphism classes of
indecomposable injective graded modules are used in an essential way. Also,
there is constructed an isomorphism of ringed spaces (Proj A,O_{Proj A}) -->
(Spec(qgr A),O_{qgr A}), where (Spec(qgr A),O_{qgr A}) is a ringed space
associated to the lattice L_{serre}(qgr A) of tensor Serre subcategories of qgr
A.Comment: some minor corrections mad
Torsion classes of finite type and spectra
Given a commutative ring R (respectively a positively graded commutative ring
A=\ps_{j\geq 0}A_j which is finitely generated as an A_0-algebra), a
bijection between the torsion classes of finite type in Mod R (respectively
tensor torsion classes of finite type in QGr A) and the set of all subsets
Y\subset Spec R (respectively Y\subset Proj A) of the form
Y=\cup_{i\in\Omega}Y_i, with Spec R\Y_i (respectively Proj A\Y_i) quasi-compact
and open for all i\in\Omega, is established. Using these bijections, there are
constructed isomorphisms of ringed spaces
(Spec R,O_R)-->(Spec(Mod R),O_{Mod R}) and
(Proj A,O_{Proj A})-->(Spec(QGr A),O_{QGr A}), where (Spec(Mod R),O_{Mod R})
and (Spec(QGr A),O_{QGr A}) are ringed spaces associated to the lattices
L_{tor}(Mod R) and L_{tor}(QGr A) of torsion classes of finite type. Also, a
bijective correspondence between the thick subcategories of perfect complexes
perf(R) and the torsion classes of finite type in Mod R is established
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